Fibonacci Using DP in DSA C - Build from Scratch

Fibonacci Using DP

📖 Scenario: You want to find the nth number in the Fibonacci sequence quickly. The Fibonacci sequence starts with 0 and 1, and each next number is the sum of the two before it.Using a smart way called Dynamic Programming (DP), you will store results of smaller problems to avoid repeating work.

🎯 Goal: Build a program that calculates the nth Fibonacci number using Dynamic Programming with an array to store intermediate results.

📋 What You'll Learn

Create an integer variable n with value 10

Create an integer array fib of size n + 1

Initialize the first two Fibonacci numbers in fib[0] and fib[1]

Use a for loop with variable i from 2 to n to fill the fib array

Print the nth Fibonacci number stored in fib[n]

💡 Why This Matters

🌍 Real World

Calculating Fibonacci numbers efficiently is useful in computer graphics, financial models, and algorithm design where repeated calculations happen.

💼 Career

Understanding Dynamic Programming and memoization is key for software engineering roles, especially in optimization and algorithm design.

1

Create the input variable and array

Create an integer variable called n and set it to 10. Then create an integer array called fib with size n + 1.

DSA C

int n = 10;
int fib[11];
// Your code here

Hint

Remember, the array size should be n + 1 to hold Fibonacci numbers from 0 to n.

2

Initialize the first two Fibonacci numbers

Set fib[0] to 0 and fib[1] to 1 to start the Fibonacci sequence.

DSA C

int n = 10;
int fib[11];
fib[0] = 0;
fib[1] = 1;
// Your code here

Hint

The first two Fibonacci numbers are always 0 and 1.

3

Fill the Fibonacci array using a loop

Use a for loop with variable i starting from 2 up to and including n. Inside the loop, set fib[i] to the sum of fib[i - 1] and fib[i - 2].

DSA C

int n = 10;
int fib[11];
fib[0] = 0;
fib[1] = 1;
for (int i = 2; i <= n; i++) {
    // Your code here
}

Hint

Each Fibonacci number is the sum of the two before it.

4

Print the nth Fibonacci number

Print the value stored in fib[n] using printf.

DSA C

int n = 10;
int fib[11];
fib[0] = 0;
fib[1] = 1;
for (int i = 2; i <= n; i++) {
    fib[i] = fib[i - 1] + fib[i - 2];
}
// Your code here

Hint

The 10th Fibonacci number is 55.

Practice

1. What is the main advantage of using Dynamic Programming (DP) to calculate Fibonacci numbers compared to a simple recursive approach?

easy

A. It stores intermediate results to avoid repeated calculations

B. It uses more memory without improving speed

C. It calculates Fibonacci numbers randomly

D. It only works for small Fibonacci numbers

Solution

1. Step 1: Understand recursion vs DP

   Simple recursion recalculates the same Fibonacci numbers many times, causing slow performance.

2. Step 2: Role of DP in Fibonacci

   DP saves previously calculated Fibonacci values in a table (memoization or tabulation), avoiding repeated work.

3. Final Answer:

   It stores intermediate results to avoid repeated calculations -> Option A

4. Quick Check:

   DP = stores results to save time [OK]

Hint: DP saves past answers to speed up Fibonacci [OK]

Common Mistakes:

• Thinking DP uses more memory without speed benefit
• Believing DP calculates numbers randomly
• Assuming DP only works for small inputs

2. Which of the following is the correct way to initialize the base cases in a bottom-up DP approach for Fibonacci in C?

easy

A. int fib[2] = {0, 0};

B. int fib[2] = {1, 1};

C. int fib[2] = {0, 1};

D. int fib[2] = {1, 0};

Solution

1. Step 1: Identify Fibonacci base cases

   Fibonacci sequence starts with fib(0) = 0 and fib(1) = 1.

2. Step 2: Match initialization with base cases

   Array fib[2] = {0, 1} correctly sets fib(0) = 0 and fib(1) = 1.

3. Final Answer:

   int fib[2] = {0, 1}; -> Option C

4. Quick Check:

   Base fib = 0,1 [OK]

Hint: Start fib with 0 and 1 for base cases [OK]

Common Mistakes:

• Swapping the order of 0 and 1
• Setting both base cases to 1
• Initializing both to zero

3. What is the output of the following C code snippet using bottom-up DP to find fib(5)?

int fib[6];
fib[0] = 0;
fib[1] = 1;
for(int i = 2; i <= 5; i++) {
    fib[i] = fib[i-1] + fib[i-2];
}
printf("%d", fib[5]);

medium

A. 5

B. 8

C. 3

D. 13

Solution

1. Step 1: Calculate Fibonacci values step-by-step

   fib[0]=0, fib[1]=1 fib[2]=1, fib[3]=2, fib[4]=3, fib[5]=5

2. Step 2: Identify fib[5]

   fib[5] is 5 after the loop completes.

3. Final Answer:

   5 -> Option A

4. Quick Check:

   fib(5) = 5 [OK]

Hint: Calculate fib stepwise from 0 and 1 [OK]

Common Mistakes:

• Confusing fib(5) with fib(6)
• Starting loop from wrong index
• Off-by-one errors in array indexing

4. Identify the error in this DP Fibonacci code snippet and select the correct fix:

int fib[10];
fib[0] = 0;
fib[1] = 1;
for(int i = 1; i < 10; i++) {
    fib[i] = fib[i-1] + fib[i-2];
}

medium

A. Initialize fib[2] before loop

B. Change loop to start from i = 2

C. Change loop condition to i <= 10

D. No error, code is correct

Solution

1. Step 1: Analyze loop start index

   The loop starts at i=1, but fib[1] is already initialized, so fib[1] gets overwritten incorrectly.

2. Step 2: Correct loop start

   Loop should start at i=2 to use fib[0] and fib[1] for calculation without overwriting base cases.

3. Final Answer:

   Change loop to start from i = 2 -> Option B

4. Quick Check:

   Loop start = 2 fixes overwrite [OK]

Hint: Start loop after base cases to avoid overwrite [OK]

Common Mistakes:

• Starting loop at 1 overwrites fib[1]
• Not initializing base cases properly
• Using wrong loop end condition

5. You want to find the 10th Fibonacci number using DP but with optimized space using only two variables instead of an array. Which code snippet correctly implements this?

hard

A. int a = 0, b = 1, c, n = 10;
for(int i = 0; i < n; i++) {
c = a + b;
a = c;
b = a;
}
printf("%d", c);

B. int a = 0, b = 1, c, n = 10;
for(int i = 1; i < n; i++) {
c = a + b;
b = a;
a = c;
}
printf("%d", a);

C. int a = 1, b = 1, c, n = 10;
for(int i = 2; i <= n; i++) {
c = a + b;
a = b;
b = c;
}
printf("%d", c);

D. int a = 0, b = 1, c, n = 10;
for(int i = 2; i <= n; i++) {
c = a + b;
a = b;
b = c;
}
printf("%d", b);

Solution

1. Step 1: Understand space optimization

   Only two variables (a, b) track last two Fibonacci numbers; c stores current sum.

2. Step 2: Check loop and updates

   Loop from 2 to n, update c = a + b, then shift a = b, b = c correctly to move forward.

3. Final Answer:

   Code snippet A correctly implements optimized Fibonacci calculation -> Option D

4. Quick Check:

   Two vars + sum update = correct [OK]

Hint: Use two vars and update in loop for space optimization [OK]

Common Mistakes:

• Swapping updates of a and b incorrectly
• Wrong loop range causing off-by-one errors
• Initializing variables incorrectly